$\begin{align}  & \int{\frac{2x+3}{{{x}^{2}}+2x+10}dx}=\int{\frac{2x+2+1}{{{x}^{2}}+2x+10}dx} \\ & \int{\frac{2x+3}{{{x}^{2}}+2x+10}dx}=\int{\frac{2x+2}{{{x}^{2}}+2x+10}dx}+\int{\frac{dx}{{{x}^{2}}+2x+10}} \\ & \int{\frac{{{x}^{2}}+2x}{{{x}^{2}}+2x+2}dx}=\log ({{x}^{2}}+2x+10)+\int{\frac{dx}{{{(x+1)}^{2}}+{{3}^{2}}}} \\ & \int{\frac{{{x}^{2}}+2x}{{{x}^{2}}+2x+2}dx}=\log ({{x}^{2}}+2x+10)+\frac{1}{3}{{\tan }^{-1}}\frac{x+1}{3}+C \\\end{align}$